I am trying to get from the abstract representation of Spinors,
as wave functions $|\Psi \rangle$ in the base of tensors products $| S_z \rangle \otimes | x \rangle$ of eigenvectors of the spin operator $\hat S_z$ and position operator $ \hat x$ to the representation in wave function form: $ \begin{pmatrix}{\Psi_+(x) \\ \Psi_-(x)} \end{pmatrix}$.

I seem to be missing something key. I would really apprechiate some help with this. Here is where i go wrong:

which then if i leave the summation over the base vectors out gives $ \begin{pmatrix}{\Psi(x) \\ \Psi(x)} \end{pmatrix}$ which is exactly not what i want. Since it is the same wave function in both components, what am i doing wrong here?

How do i get different wave functions for the different components using the dirace formalism? I would really be glad about some Tipps.